马兰·梅森Marin Mersenne

马兰·梅森于1588年9月8日出生在缅因省小镇奥伊泽的一个工人阶级家庭，并在同一天受洗。从很小的时候起，他就表现出虔诚和渴望学习的迹象。因此，尽管经济状况不佳，梅森的父母还是把他送到勒芒学院，在那里他上语法课。后来，在十六岁时，梅森要求去拉弗莱什新成立的耶稣会学校，这所学校是作为模范学校建立的，为所有儿童谋福利，不论其父母的经济状况如何。事实证明，比梅森小八岁的勒内·笛卡儿也就读于同一所学校，尽管人们认为他们直到很久以后才成为朋友。

梅森的父亲希望儿子在教会中谋职。然而，梅森热爱学习，并表明他已准备好承担世俗责任，决定去巴黎深造。他前往巴黎，途中住在最小兄弟会的一座修道院。这段经历深深启发了梅森，以至于他同意如果有一天决定过修道生活，就加入他们的修会。到达巴黎后，他在法兰西皇家学院学习，继续接受哲学教育，并在索邦大学听神学课程，还获得了哲学硕士学位。他于1611年完成学业，由于受过优越的教育，他意识到自己现在已经准备好过修道院的平静而勤奋的生活。

最小兄弟会由圣弗朗西斯·德·保拉于1436年创立，此时正蓬勃发展。他们认为自己是世上所有宗教中最小的（minimi），致力于祈祷、学习和学术。他们穿着由粗糙黑色羊毛制成的会衣，袖子宽大，用一根细黑绳束腰（如梅森的肖像中所见）。查理八世将该修会引入法国，由于修士们极为简朴，他们被称为“les bons hommes”。法国大革命后，该修会人数大幅减少，如今在意大利仅存几座修道院。梅森于1611年7月16日加入该修会，并在尼容和莫修道院经过两个半月的试用期后，于1612年7月在巴黎被任命为神父。他的第一个职务是1614年被派往讷韦尔的修道院，在那里向修会年轻成员教授哲学和神学。事实上，他的一个学生希拉里翁·德·科斯特后来成为他的知己和传记作者。正是在他生命的这一时期，人们认为他发现了摆线——一条几何曲线。

任教两年后，梅森被选为巴黎皇家广场修道院的院长，除了短暂的旅行外，他一直留在那里，直到1648年去世。据信，教会在他一生的大部分时间里都支持他，尽管在晚年，一位同修雅克·阿莱在经济上帮助他，并允许他使用自己的图书馆。从他在巴黎开始，数学问题就在他的生活中扮演了重要角色。他很早就与巴黎的重要学者建立了联系，经常与他们见面，交流思想，讨论项目。最小兄弟会意识到他能提供的最大服务是通过他的著作，因此从未对他提出更多要求。

1623年，他发表了他的前两篇论文，包括在法国反对无神论和怀疑论的研究；L'usage de la raison Ⓣ（理性的运用）和L'analyse de la vie spirituelle Ⓣ（精神生活的分析）。继续他的神学写作后，他当时想反驳魔法，然而一位同修指出这不合适，导致他出版了Quaestiones celeberrime in genesim Ⓣ（关于《创世记》的重要问题），其中包括对《圣经》中魔术师的反对。这本书包含《圣经》前六章文本的1900栏。正是由于这一出版物，1624年9月他回到巴黎时，遇到了皮埃尔·伽桑狄，后者曾被要求评论梅森的结果，后来成为他最亲密的朋友。

此时法国正经历一段反巫术时期，驱逐任何巫师。L'impiété des deistes Ⓣ（自然神论者的不敬）以法文写成，面向法国公众，以便他们能够阅读并理解正在发生的事情。正是在这一时期，梅森 开始思考针对 勒内·笛卡儿 和 伽利略 的神学批评。事实上，正如 Garber 在 [16] 中指出的，梅森 对 伽利略 的态度在若干年间发生了变化：-

梅森 是17世纪30年代和40年代巴黎新的自然数学方法的核心人物。在思想上，他是该纲领最热情的实践者之一，并在那些重要的几十年里出版了许多有影响力的著作。但 梅森 的职业生涯起步方式颇为不同。在17世纪20年代初，梅森 在巴黎主要以宗教题材作家而闻名，并且是 亚里士多德 的坚定捍卫者，反对那些想用新哲学取代他的人的攻击。……在17世纪20年代初，梅森 将 伽利略 列为自然哲学中应被拒绝的创新者之一。然而，到17世纪30年代初，不到十年之后，梅森 已成为 伽利略 最热忱的支持者之一。

梅森 开始意识到，除了宗教之外，真正让他感兴趣的是科学。数学是他研究最深入的领域，他相信没有数学就不可能有任何科学。他对数学始终持哲学态度，并相信科学的事业就是上帝的事业，见 [5]。因此，在 La vérité des sciences Ⓣ（科学的真理）中，他通过许多伟大发现证明了人类心智的价值。大约在这个时候，梅森 开始成为所有欧洲学者的协调者。从1623年起，他开始精心挑选学者，这些学者在他的巴黎修道院聚会，或从欧洲各地甚至远至君士坦丁堡和特兰西瓦尼亚（今罗马尼亚）与他通信。他的常客或通信者包括 Peiresc、皮埃尔·伽桑狄、勒内·笛卡儿、罗贝瓦尔、Beeckman、J B van Helmont、皮埃尔·德·费马、托马斯·霍布斯、埃蒂安·帕斯卡尔 以及他的儿子 布莱兹·帕斯卡。他组织来自欧洲各地的学者会议，会上他们阅读和评论国内外的科学论文，与其他学者交换联系方式，并计划和讨论实验及其他工作。这后来被称为 Académie Parisiensis，在朋友中有时被称为 Académie Mersenne。它显然是当时最具资源的研究中心之一，每周在成员家中聚会，后来由于他健康状况不佳，改在 梅森 的修道院小室中聚会。梅森 的通信者名单不断增加，梅森 本人也毫不犹豫地前往欧洲各地与学者会面。

梅森 对音乐有浓厚兴趣，并花费大量时间研究声学和声速。1627 年，他出版了他最著名的著作之一，L'harmonie universelle Ⓣ（普遍和谐）。在这部著作中，他首次发表了关于振动弦的定律：在改变其中一个量时，若其他条件保持不变，其频率与张力的平方根成正比，与长度、直径以及弦的比重平方根成反比。梅森 已经开始鼓励他人的才能，并帮助他们与其他学者分享自己的想法和成果。当 罗贝瓦尔 抵达巴黎，加入 梅森 的学者圈子后，他的才能很快被 梅森 认可，后者鼓励他研究摆线。

1627 年至 1634 年之间是 梅森 一生中的过渡时期。在此期间，他于 1629 年至 1630 年间前往荷兰数月。他的主要原因是借助温泉疗养来寻求治疗疾病，但他利用这个机会拜访了周边地区的学者。当 Questions inouyes Ⓣ（未解答的问题）和 Questions harmoniques Ⓣ（和声问题）于 1634 年印行时，他自上次出版以来七年中写作上更为成熟这一点变得明显。1644 年 10 月，梅森 前往普罗旺斯和意大利，在那里他从 埃万杰利斯塔·托里拆利 那里得知了气压计实验。回到巴黎后，他报告了这一消息，以鼓励法国学者也进行这些实验。

在他一生中，梅森 通过将许多潜在科学家引向正确方向，并就下一步该做什么向其中一些人提供建议，帮助了他们。他成为 克里斯蒂安·惠更斯 的榜样，梅森 将 克里斯蒂安·惠更斯 收于翼下，并通过鼓励性的书信激发了 克里斯蒂安·惠更斯 的 Theory of Music。克里斯蒂安·惠更斯 曾打算于 1646 年搬到巴黎，以便靠近 梅森，使他们能够更容易地相互联系，然而 克里斯蒂安·惠更斯 直到 梅森 去世数年后才搬去，因此他们从未见面。

伽利略 也必须感谢 梅森 使他的著作在意大利之外为人所知。梅森 坚持出版 伽利略 的著作，若没有这一点，伽利略 的思想可能永远不会如此广为人知。梅森 在晚年仍继续旅行，于 1646 年启程前往波尔多。在那里，他遇到了 Pierre Trichet，并帮助后者崭露头角。波尔多和吉耶讷地区科学生活的成功——后来形成了 Académie Royale des Sciences——在很大程度上归功于 梅森 能够提供的建议和经验。他于 1647 年返回巴黎。

梅森在1648年7月去探望勒内·笛卡儿后病倒了，不幸的是，他的健康再也没有好转。有人建议他在水中掺酒以助康复，然而最小兄弟会成员不饮酒。他的肺部有脓肿，但外科医生未能找到。梅森本人指出，他要求做的切口位置太低。皮埃尔·伽桑狄在梅森患病期间一直陪伴着他，直到他于1648年9月1日在巴黎去世，距离他60岁生日仅8天。他从未放弃毕生推进科学的愿望。他甚至立下遗嘱，要求将遗体用于生物学研究。

梅森去世后，在他的牢房里发现了来自78位不同通信者的信件，包括皮埃尔·德·费马、克里斯蒂安·惠更斯、约翰·佩尔、伽利略和埃万杰利斯塔·托里拆利。还在他的牢房里发现了几件物理仪器，并找回了梅森的大量藏书，其中L'optique et la catoptriqueⓉ（光学与反射光学）于1651年出版。在这部出版物中，插入了罗贝瓦尔的一篇文本。后来，他寄出和收到的与其他学者的所有信件被汇集起来，分几卷出版。这些信件读起来像是一部17世纪初力学的国际评论。梅森了解当时正在进行的所有科学，所有科学家在做什么，他只希望他们都能共同致力于推进科学。

梅森研究摆线数年，并在Quaestiones in GenesimⓉ（《创世记问题》）（1623年）、Synopsis mathematicaⓉ（《数学概要》）（1626年）和Questions inouyesⓉ（《难以置信的问题》）（1634年）中引用了他的研究。他将摆线定义为距半径为$a$的圆的圆心距离为h的点的轨迹，该圆沿直线滚动。他陈述了显而易见的性质，包括基线长度等于滚动圆的周长。我们注意到梅森将摆线称为“轮盘线”，但后来采用了摆线这一术语。他试图通过积分求曲线下的面积，但失败了，于是他将问题提交给罗贝瓦尔。1638年，他宣布罗贝瓦尔确实求出了摆线下的面积。

梅森 的名字今天最为人记住的是 梅森 primes。他试图找到一个能表示所有素数的公式，但尽管他失败了，他对形如

$2^{p} - 1$，$p$ 素数

的数的工作在大素数的研究中一直引人关注。很容易证明，如果数 $n = 2^{p} - 1$ 是素数，那么 $p$ 必定是素数。1644年，梅森 声称如果 $p$ = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 和 257，则 $n$ 是素数，但对于小于257的另外44个素数 $p$，composite。

多年来，人们发现梅森在形如$2^{p} - 1$的素数中，当$p$小于或等于257时，有5个是错误的（他声称两个不会导致素数（67和257），并遗漏了3个确实导致素数的：61、89、107）。Drake [13]试图既理解梅森关于这些素数的工作来源，也试图确定所使用的规则。他提出福兰尼可可能是来源，并建议这些错误可能是印刷者的误印。Drake重建了梅森关于指数的规则，即它们与$2^{n}$的值相差不超过一，或与2的$2^{n}$次幂的值相差不超过三。

梅森进行了实验，以检验伽利略的落体运动定律。1634年，他展示了从147、108和48英尺高度测量落体加速度时获得的结果。这些结果证实了伽利略在1632年的Dialogue on the two chief world systems中发表的时间平方定律，但也引发了关于数值数据的疑问。他试图解决的一个问题是，加速度是如伽利略所主张的那样连续，还是如勒内·笛卡儿所认为的那样不连续。梅森认为伽利略关于落体经历无限速度程度的假设与对加速度的真正机械论解释不相容。这些思想在[24]和[25]中有详细讨论。

在他的一些非数学著作中，梅森考察了排列与组合。他给出了计算组合数或排列数的实用规则，解决了求有重复或无重复排列数的问题，并给出了一个构造变位词的例子。然而，他研究组合分析的主要原因是优化音乐创作，正如他在The book on the art of singing well中所解释的，The book on the art of singing well是Harmonie universelleⓉ（《普遍和谐》）（1636年）的第六卷。在巴黎国家图书馆保存的一份未发表手稿中，他给出了8个音符的40320种排列。

在他生命的最后四年里，梅森花了大量时间研究气压计。布莱兹·帕斯卡已经证明空气并非没有重量，而正是梅森发现空气的密度大约是水的$\large\frac{1}{19}\normalsize$分之一。他通过De Verdus的几封信得知了气压计实验，该实验用一个约3英尺长、一端密封并装满纯水银的玻璃管进行，但直到1644年10月，当他访问意大利的埃万杰利斯塔·托里拆利时，他才亲眼看到实验的进行。埃万杰利斯塔·托里拆利用空气的压力来解释为什么水银会沿玻璃管上升。梅森怀疑空气压力是否真的支撑着水银，回来后试图重做实验，但没有必要的设备。梅森向他在巴黎的儿子埃蒂安·帕斯卡尔、布莱兹·帕斯卡、亚历克西·泰雷兹·珀蒂、罗贝瓦尔等人解释了这个问题。关于1647年是谁最初提出用托里拆利管和山进行实验（后来被称为多姆山实验），存在一些混淆。当然，梅森曾向克里斯蒂安·惠更斯和Le Tenneur简要介绍过，但直到1648年梅森去世三周后，这些实验才得以进行。实验包括在多姆山脚下和山顶收集结果。他们测试了山顶水银柱中的水银水平是否比山脚下低。如果这被证明是真的，他们意识到这只会是由于空气的压力。最终进行实验的Perier确实发现水银水平有显著差异，从而得出正确的结论：空气压力支撑着它。

一个有趣的问题是，梅森是如何在教会（他是教会的一名虔诚成员）着手阻止此类讨论的时代，设法自由地追求他的科学思想的。这个话题在[18]中有详细讨论，其中Hine写道：-

在十七世纪上半叶，关于哥白尼假说的争论已经超出了天文学家的范围，并激起了如此多的争议，以至于教会决定介入。1616年，一个神学审查机构得出结论，地球运动的观点在哲学上是错误的，并且与圣经相冲突，于是它暂停了尼古拉·哥白尼的著作，直到其被修正。历史学家通常认为，这一决定以及随后对伽利略的谴责产生了如此毁灭性的影响，以至于天主教国家的科学进步大大受阻。然而，梅森的态度——他既是一个宗教修会的忠实成员，又是法国科学发展的核心人物——并不支持这样的结论。对梅森对哥白尼主义反应的考察表明，无论教会的决定多么令人不安，至少在天主教国家，仍然有可能研究哥白尼的思想并发现它们有用，尽管有一些保留意见。梅森受到了教会此类决定的影响，但影响比人们可能设想的要小。

Marin Mersenne was born into a working class family in the small town of Oizé in the province of Maine on 8 September 1588 and was baptised on the same day. From an early age he showed signs of devotion and eagerness to study. So, despite their financial situation, Marin's parents sent him to the Collège du Mans where he took grammar classes. Later, at the age of sixteen, Mersenne asked to go to the newly established Jesuit School in La Flèche which had been set up as a model school for the benefit of all children regardless of their parents' financial situation. It turns out that Descartes, who was eight years younger than Mersenne, was enrolled at the same school although they are not thought to have become friends until much later.

Mersenne's father wanted his son to have a career in the Church. Mersenne, however, was devoted to study, which he loved, and, showing that he was ready for responsibilities of the world, had decided to further his education in Paris. He left for Paris staying en route at a convent of the Minims. This experience so inspired Mersenne that he agreed to join their Order if one day he decided to lead a monastic life. After reaching Paris he studied at the Collège Royale du France, continuing there his education in philosophy and also attending classes in theology at the Sorbonne where he also obtained the degree of Magister Atrium in Philosophy. He finished his studies in 1611 and, having had a privileged education, realised that he was now ready for the calm and studious life of a monastery.

The Order of the Minims, having been set up by St Francis of Paula in 1436, was thriving at this time. They believed they were the least (minimi) of all the religions on earth, and devoted themselves to prayer, study, and scholarship. They wore a habit made of coarse black wool with broad sleeves and girded by a thin black cord (as seen in the portraits of Mersenne). Charles VIII introduced the Order into France and, due to their great simplicity, the monks were named 'les bons hommes'. After the French Revolution the Order dwindled considerably in number and today there exists only a few convents in Italy. Mersenne entered the Order on 16 July 1611, and was ordained a priest in Paris in July 1612 after a two and a half month probationary period in the monasteries at Nigeon and Meaux. His first posting was in 1614 to the monastery in Nevers where he taught philosophy and theology to the younger members of the community. In fact one of his students, Hilarion de Coste, later became his confidant and biographer. It was during this period of his life that he is thought to have discovered the cycloid - a geometric curve.

After two years teaching Mersenne was elected superior of the Place Royale monastery in Paris where he remained, except for brief journeys, until his death in 1648. It is believed that the Church supported him for most of his life, although in later years a fellow monk, Jacques Hallé, helped out with money and granted him access to his library. From the beginning of his time in Paris, mathematical problems played an important role in his life. Very early on he had links with important scholars in Paris whom he met often, exchanging ideas and discussing projects. The Minims realised that the biggest service he could give was through his books and they never asked any more of him.

In 1623 he published his first two papers consisting of studies against atheism and scepticism in France; L'usage de la raison Ⓣ and L'analyse de la vie spirituelle Ⓣ. Continuing his theological writing he had then wanted to disprove magic, however a fellow monk pointed out that it was not appropriate, leading to his publication of Quaestiones celeberrime in genesim Ⓣ that includes the disapproval of magicians in the Scriptures. This book contains 1900 columns of text from the Bible in its first six chapters. It was because of this publication that, in September 1624 when he returned to Paris, he met Gassendi who had been asked to comment on Mersenne's results, and later became his closest friend.

At this time France was going through a period of anti-witchcraft, expelling any sorcerers. L'impiété des deistes Ⓣ, in French, was aimed at the French public so that they might read and understand what was happening. It was during this time that Mersenne started to think about the theological criticism directed against Descartes and Galileo. In fact Mersenne's attitude to Galileo changed over a number of years as Garber points out in [16]:-

Marin Mersenne was central to the new mathematical approach to nature in Paris in the 1630s and 1640s. Intellectually, he was one of the most enthusiastic practitioners of that program, and published a number of influential books in those important decades. But Mersenne started his career in a rather different way. In the early 1620s, Mersenne was known in Paris primarily as a writer on religious topics, and a staunch defender of Aristotle against attacks by those who would replace him by a new philosophy. ... In the early 1620s, Mersenne listed Galileo among the innovators in natural philosophy whose views should be rejected. However, by the early 1630s, less than a decade later, Mersenne had become one of Galileo's most ardent supporters.

Mersenne was beginning to realise that alongside religion it was science that really interested him. Mathematics was the area he studied in greatest depth, believing that without it no science was possible. He always had a philosophical approach to mathematics and believed that the cause of the sciences is the cause of God, see [5]. So, in La vérité des sciences Ⓣ he proved, via many great discoveries, the value of the human mind. It was around this time that Mersenne started to become a coordinator for all European scholars. From 1623 he began to make a careful selection of savants who met at his convent in Paris or corresponded with him from all across Europe and even from as far afield as Constantinople and Transylvania (present-day Romania). His regular visitors, or correspondents, included Peiresc, Gassendi, Descartes, Roberval, Beeckman, J B van Helmont, Fermat, Hobbes, Étienne Pascal, and his son Blaise Pascal. He set up meetings of scholars from around Europe during which they would read and review scientific papers, both national and international, exchange contacts with other scholars and plan and discuss experiments and other work. This came to be known as the Académie Parisiensis and sometimes among friends as the Académie Mersenne. It was notably one of most resourceful centres of research at that time, meeting weekly at members' houses and later in Mersenne's cell due to his weakened health. The list of Mersenne's correspondents kept increasing and Mersenne himself did not hesitate to travel to meetings with scholars all around Europe.

Mersenne had a strong interest in music and spent a lot of time researching acoustics and the speed of sound. In 1627 he published one of his most famous works, L'harmonie universelle Ⓣ. In this work he was the first to publish the laws relating to the vibrating string: its frequency is proportional to the square root of the tension, and inversely proportional to the length, to the diameter and to the square root of the specific weight of the string, provided all other conditions remain the same when one of these quantities is altered. Mersenne had already started encouraging the talents of others and helped them to share their ideas and results with other scholars. When Roberval arrived in Paris, after joining Mersenne's circle of scholars, his talent was soon recognised by Mersenne who encouraged him to work on the cycloid.

The period between 1627 and 1634 was a transitional period in Mersenne's life. During this time he travelled to Holland for several months between 1629 and 1630. His main reason was to seek a cure for an illness with the help of spa water but he used the opportunity to visit scholars in the surrounding areas. The greater maturity in his writing in the seven years since his last publication became apparent when Questions inouyes Ⓣ and Questions harmoniques Ⓣ were printed in 1634. In October 1644 Mersenne travelled to Provence and Italy where he learnt of the barometer experiment from Torricelli. On his return to Paris, he reported this news to encourage French scholars to carry out the experiments too.

Throughout his lifetime Mersenne helped many potential scientists by steering them in the right direction and advising some on the next step to take. He became a role model for Huygens whom Mersenne took under his wing and through his encouraging letters inspired Huygens' Theory of Music. Huygens had intended to move to Paris in 1646 to be near Mersenne in order to enable them to contact each other more easily, however Huygens did not move until several years after Mersenne had died so they never met.

Galileo also has to be grateful to Mersenne for making his work known outside Italy. Mersenne insisted on publishing Galileo's work and without this Galileo's ideas might never have become as widely known. Continuing his travels into his old age, in 1646 Mersenne set off on a trip to Bordeaux. There he met Pierre Trichet whom he helped make his mark. The success of the scientific life over in Bordeaux and Guyenne, which later formed the Académie Royale des Sciences, was largely due to the advice and experience Mersenne was able to offer. He returned to Paris in 1647.

Mersenne fell ill after his visit to see Descartes in July 1648 and, unfortunately, his health never improved. He was advised to mix wine with his water to help him get better, however Minims do not drink wine. He had an abscess on the lung but the surgeon was unable to find it. Mersenne himself pointed out that the incision, which he asked for, had been attempted too low. Gassendi was there for Mersenne throughout his illness and remained with him until his death on 1 September 1648 in Paris, just 8 days from his 60th birthday. He never gave up his life-long desire to advance science. He even asked, in his will, that his body be used for biological research.

After Mersenne's death, letters in his cell were found from 78 different correspondents including Fermat, Huygens, Pell, Galileo and Torricelli. Also several physics instruments were found in his cell and a lot of Mersenne's library was retrieved from which L'optique et la catoptrique Ⓣ was published in 1651. Inside this publication one of Roberval's texts was inserted. Later all the letters he sent and received from other scholars were accumulated and published in several volumes. These letters read like an international review of mechanics in the early 17th century. Mersenne was aware of all the science that was going on, what all the scientists were doing, and only wanted for them all to work together in advancing science.

Mersenne studied the cycloid for several years quoting his research in Quaestiones in Genesim Ⓣ (1623), Synopsis mathematica Ⓣ (1626) and Questions inouyes Ⓣ (1634). He gave the definition of a cycloid as the locus of a point at distance h from the centre of a circle of radius $a$, that rolls along a straight line. He stated the obvious properties including the length of the base line equals the circumference of the rolling circle. We note that Mersenne referred to the cycloid as the 'roulette' but the term cycloid was adopted later. He attempted to find the area under the curve by integration but having failed, so he put the question to Roberval. In 1638 he announced that Roberval had indeed found the area under the cycloid.

Mersenne's name is best remembered today for Mersenne primes. He tried to find a formula that would represent all primes but, although he failed in this, his work on numbers of the form

$2^{p} - 1$, $p$ prime

has been of continuing interest in the investigation of large primes. It is easy to prove that if the number $n = 2^{p} - 1$ is prime then $p$ must be a prime. In 1644 Mersenne claimed that $n$ is prime if $p$ = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 but composite for the other 44 primes $p$ smaller than 257.

Over the years it has been found that Mersenne was wrong about 5 of the primes of the form $2^{p} - 1$ where $p$ is less than or equal to 257 (he claimed two that did not lead to a prime (67 and 257) and missed 3 that did: 61, 89, 107). Drake [13] has tried to both understand the source of Mersenne's work on these primes, and also to try to determine the rule that was being used. He suggests Frenicle de Bessy may be the source and also suggests that the errors might be misprints by the printer. Drake reconstructs Mersenne's rule for exponents as that they must differ by not more than one from a value of $2^{n}$ or by not more than three from a value of 2 to the power $2^{n}$.

Mersenne undertook experiments to test Galileo's law of motion for falling bodies. In 1634 he presented the results that he had obtained when measuring the acceleration of falling bodies from heights of 147, 108 and 48 feet. These confirmed the time-squared law that Galileo had published in his Dialogue on the two chief world systems of 1632 but they also raised questions about the numerical data. One problem he tried to solve was whether acceleration was continuous as Galileo maintained or discontinuous as Descartes believed. Mersenne thought Galileo's assumption that a falling body passes through infinite degrees of speed was incompatible with a genuinely mechanistic explanation of acceleration. These ideas are discussed in detail in [24] and [25].

In some of his non-mathematical works Mersenne looks at permutations and combinations. He states practical rules for calculating the number of combinations or permutations, solving the problem of finding the number of permutations with or without repetitions and gives an example the making of anagrams. His main reason to study combinatorial analysis was, however, to optimise musical composition as he explains in The book on the art of singing well which is Book Six of Harmonie universelle Ⓣ (1636). In an unpublished manuscript preserved in the Bibliothèque Nationale at Paris he gave the 40320 permutations of 8 notes.

During the final four years of his life, Mersenne spent a lot of time investigating the barometer. Pascal had already proved that air was not weightless and it was Mersenne who found the density of air to be approximately $\large\frac{1}{19}\normalsize$th that of water. He was informed of the barometer experiment, consisting of a glass tube about 3 feet long sealed at one end and filled with pure mercury, through several letters from De Verdus but it was not until October 1644, when he visited Torricelli in Italy, that he saw the experiment carried out. Torricelli used the pressure of the air to explain why the mercury moved up the glass tube. Mersenne was doubtful that the air pressure actually supported the mercury and on his return attempted to re-do the experiment but did not have the necessary equipment. Mersenne explained the problem to Étienne Pascal, his son Blaise Pascal, Petit, Roberval, and others in Paris. There is some confusion as to who, in 1647, initially suggested the experiments with the Torricellian tube and the mountain, later to be called the Puy de Dome experiments. Certainly Mersenne had briefed both Huygens and Le Tenneur but it was not until three weeks after Mersenne's death in 1648 that these experiments were carried out. They consisted of collecting results both at the foot of the Puy de Dome and at the summit. Tests were made as to whether the level of mercury in the column was lower when at the top of the mountain than it was at the bottom. If this had proved to be true, they realised that this would be due to the pressure of the air alone. Perier, who finally conducted the experiments, did indeed find that there was a significant difference in the level of the mercury hence drawing the correct conclusion that the air pressure was supporting it.

An interesting question is how Mersenne managed to pursue his scientific ideas freely at a time when the Church (of which he was a devoted member) moved to prevent such discussion. This topic is considered in detail in [18] where Hine writes:-

During the first half of the seventeenth century, debate over the Copernican hypothesis had spread beyond the ranks of astronomers and had stirred up so much controversy that the Church decided to intervene. In 1616 a theological examining body concluded that the idea of the earth's motion was philosophically false and in conflict with the Scriptures, and it suspended Copernicus's book until corrected. Historians have generally assumed that this decision and the subsequent condemnation of Galileo had such a devastating effect that scientific progress in Catholic countries was greatly retarded. However, the attitude of Mersenne, who was both a faithful member of a religious order and a central figure in the development of French science, does not support such a conclusion. An examination of Mersenne's reaction to Copernicanism indicates that no matter how disturbing the Church's decision, it was still possible, at least in France, to study Copernican ideas and to find them useful, despite some reservations. Mersenne was affected by such decisions of the Church, but less so than one might suppose.